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lecture_32 (dragged) - MA 36600 LECTURE NOTES: MONDAY,...

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Unformatted text preview: MA 36600 LECTURE NOTES: MONDAY, APRIL 13 Basic Theory of Systems of First Order Linear Equations Solutions to Systems of Linear Differential Equations. We continue to focus on a system of first order differential equations in the form x￿ 1 x￿ 2 x￿ m = = . . . p11 (t) x1 p21 (t) x1 = pm1 (t) x1 + + p12 (t) x2 p22 (t) x2 + ··· + ··· . . . + pm2 (t) x2 + ··· + + p1n (t) xn p2n (t) xn + pmn (t) xn + + g1 (t) g2 (t) + gm (t) Such a system contains m equations and n variables. When m = n, we can express this as d x = P(t) x + g(t). dt We focus on a few special cases to gain intuition about the solutions of this system. Example #1. Consider first when m = n = 1. The first order equation x￿ = p(t) x + g (t) can be solved using integrating factors: ￿ ￿t ￿ µ(t) = exp − p(τ ) dτ 1 x(t) = µ(t) =⇒ ￿￿ t µ(τ ) g (τ ) dτ + C . Hence the general solution is in the form x(t) = C x(1) (t) + x(p) (t), where ￿￿ t ￿ 1 d (1) x(1) (t) = = exp p(τ ) dτ =⇒ x = p(t) x(1) µ(t) dt is a solution to the homogeneous equation, and (p) x 1 (t) = µ(t) ￿ t ￿ µ(τ ) g (τ ) dτ is a particular solution to the nonhomogeneous equation. Example #2. Consider now when m = n = 2. The equation x￿￿ = q (t) x + p(t) x￿ + g (t) can be transformed into a system of first order equations through the substitution ￿ ￿ ￿ ￿ ￿ ￿ d x 0 1 0 x= =⇒ x= x+ . x￿ q (t) p(t) g (t) dt Say that we have three functions x1 = x1 (t), x2 = x2 (t), and X = X (t) such that i. x1 and x2 are solutions to the homogeneous equation x￿￿ − p(t) x￿ − q (t) x = 0. ii. Their Wronskian is nonzero, in terms of the function ￿ ￿ ￿￿ t ￿ ￿ ￿ x1 (t) x2 (t) W x1 , x2 (t) = x1 (t) x￿ (t) − x￿ (t) x2 (t) = det = C exp p(τ ) dτ . 2 1 x￿ (t) x￿ (t) 1 2 iii. X is a particular solution to the nonhomogeneous equation X ￿￿ − p(t) X ￿ − q (t) X = g (t). 1 ...
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